Difference between revisions of "HW 3.3 - Xiaodian Xie ECE301 Summer2009" - Rhea

Latest revision as of 18:03, 8 July 2009

Derivation of Linearity for CT signals by Xiaodian Xie

Suppose z(t) = {ax(t)+by(t)}, then the fourier transform of z is z(w)=d^(-1)/dt((ax(t)+by(t))*exp(-jwt))

so z(w)=d^(-1)/dt(ax(t)*exp(-jwt)))+d^(-1)/dt(by(t)*exp(-jwt)))

So z(w)=ax(w)+by(w)

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 Derivation of Linearity for CT signals by Xiaodian Xie
-Suppose z(t) = {ax(t)+by(t)}, then the fourier transform of z is z(w)=\int\limits_{-\infty}^{\infty}(ax(t)+by(t))e^{(-\jmath wt)}dt
-z(w)=\int\limits_{-\infty}^{\infty}ax(t)e^{(-\jmath wt)}dt+\int\limits_{-\infty}^{\infty}by(t)e^{(-\jmath wt)}dt
-z(w)=a\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt+b\int\limits_{-\infty}^{\infty}y(t)e^{(-\jmath wt)}dt
-Because x(w)=\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt (Same for Y(w));So we can say that z(w)=a*x(w)+b*y(w)
+Suppose z(t) = {ax(t)+by(t)}, then the fourier transform of z is z(w)=d^(-1)/dt((ax(t)+by(t))*exp(-jwt))
+so z(w)=d^(-1)/dt(ax(t)*exp(-jwt)))+d^(-1)/dt(by(t)*exp(-jwt)))
+So z(w)=ax(w)+by(w)